An Action for and Hydrodynamics from the improved Large D membrane

Abstract

It has recently been demonstrated that black hole dynamics at large D is dual to the motion of a probe membrane propagating in the background of a spacetime that solves Einstein's equations. The equation of motion of this membrane is determined by the membrane stress tensor. In this paper we `improve' the membrane stress tensor derived in earlier work to ensure that it defines consistent probe membrane dynamics even at finite $D$ while reducing to previous results at large D. Our improved stress tensor is the sum of a Brown York term and a fluid energy momentum tensor. The fluid has an unusual equation of state; its pressure is nontrivial but its energy density vanishes. We demonstrate that all stationary solutions of our membrane equations are produced by the extremisation of an action functional of the membrane shape. Our action is an offshell generalization of the membrane's thermodynamical partition function. We demonstrate that the thermodynamics of static spherical membranes in flat space and global AdS space exactly reproduces the thermodynamics of the dual Schwarzschild black holes even at finite D. We study the long wavelength dynamics of membranes in AdS space, and demonstrate that the boundary `shadow' of this membrane dynamics is boundary hydrodynamics with with a definite constitutive relation. We determine the explicit form of shadow dual boundary stress tensor upto second order in derivatives of the boundary temperature and velocity, and verify that this stress tensor agrees exactly with the fluid gravity stress tensor to first order in derivatives, but deviates from the later at second order and finite D.

64 pages, v2: footnote added, some typos corrected

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• Black holes in presence of cosmological constant: Second order in 1/D
• Holographic Bjorken Flow at Large-D
• Stationary Solutions from the Large D Membrane Paradigm
• Large D gravity and charged membrane dynamics with nonzero cosmological constant
• Holographic Turbulence in Einstein-Gauss-Bonnet Gravity at Large $D$
• Entropy production and entropic attractors in black hole fusion and fission
• The large D Membrane Paradigm For Einstein-Gauss-Bonnet Gravity
• A leading-order comparison between fluid-gravity and membrane-gravity dualities
• Large D membrane for Higher Derivative Gravity and Black Hole Second Law
• Fluid-gravity and membrane-gravity dualities - Comparison at subleading orders
• The large $D$ membrane paradigm for general four-derivative theory of gravity with a cosmological constant
• The large D effective theory of black strings in AdS
• Black Rings in Large $D$ Membrane Paradigm at the First Order
• Black hole interactions at large $D$: brane blobology
• General Theory of Large D Membranes Consistent with Second Law of Thermodynamics
• Study of Semiclassical Instability of the Schwarzschild AdS Black Hole in the Large $D$ Limit
• Stress Tensor for Large-$D$ membrane at subleading orders
• Schwarzschild-Tangherlini quasinormal modes at large $D$ revisited